HOMEOMORPHIC CONTINUOUS CURVES IN 2-SPACEARE ISOTOPIC IN 3-SPACE BY W. K. MASON« §1 1. Introduction. The following question appears in [12, p. 57]. If Kx and K2 are homeomorphic compact continua in F2, is there an isotopy (or even a homeo- morphism) of E3 onto F3 which carries Kx onto F2? (See also [5, p. 230].) At present the answer to this question is unknown. If F3 is replaced by F4 an affirmative answer has been obtained by Klee [6, Theorem 3.3, p. 36]. Related results for complexes in higher dimensional hyperplanes have been obtained by Bing and Kister [2]. Conditions under which a homeomorphism between planar continuous curves can be extended to F2 have been obtained by Gehman [4], and Adkisson and MacLane [1]. Conditions under which a continuous curve can be embedded in F2 have been obtained by Claytor [3]. The following result provides an affirmative answer to the above question when Kx and K2 are continuous curves. Extension Theorem. Suppose that S and S' are continuous curves in E2 and g is a homeomorphism of S onto S'. Then there is a homeomorphism 77 of E3 onto E3 such that (a) H=g on S, and (b) 77 is realizable by an isotopy. Because of space limitations this article omits certain parts of the proof. Com- plete details will be found in [7]. 2. Outline of proof. The proof of the extension theorem is divided into two parts. First, the theorem is established for continuous curves without separating points. Next we show that by using an "enlarging process", the general case can be reduced to this special case. If S and S' contain no separating points the general idea of the proof is as follows: there is a planar disk D containing S' such that the boundary of D is a subset of S'. We construct subdivisions Cx, C2, C3,... of D into disks such that, for each z, (a) the interior of every element (disk) in C¡ is either a complementary domain of 5" or has diameter no greater than 1/2', and (b) the boundary of every disk in C( Received by the editors April 10, 1968. O The results presented in this paper are a part of the author's Ph.D. Thesis at the University of Wisconsin, written under the direction of Joseph Martin. Research partially supported by NSF GP-7085. 269 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 270 W. K. MASON [August is a subset of S'. We construct homeomorphisms Hx, H2, H3,... such that, for each / and each xeS, (a) Hx(x) and g(x) belong to the same disk of C¡, and (b) if g(x) is on the boundary of a disk in Ch then Ht(x)=g(x). The limit of the 77('s is the required homeomorphism 77. Now assume that S contains separating points. The general idea of the second part of the proof is as follows: we may suppose that S contains a nondegenerate cyclic element Cx. We enlarge Ci and g(Cx) by attaching arcs to S and S' until we obtain continuous curves S2=>S and S'Z=>S', and cyclic elements C2=>CXand C2=>g(Ci) of S2 and S'2 respectively such that (a) the components of S2 —C2 and of S'2— C2 have small maximum diameter, (b) g: S-^-S' may be extended to a homeomorphism g2 : S2 -»■S2, and (c) there are homeomorphisms 772,H2 : E3 -»■E3 such that H2(S2) and 772(S2) are subsets of A2. We continue this process, adding arcs of smaller and smaller diameter to 5" and S', and thereby constructing larger and larger cyclic elements. Finally, we obtain continuous curves Sm =>S and S'a =>S' such that (a) S„ and S'*,contain no separating points, (b) there is a homeomorphism gm : S1«,-*■ S'x which extends g: S-»- 5", and (c) there are homeomorphisms 77, 77' : A3 -»- A3 such that H(Sœ) and H'(S'o,) are subsets of A2. §H Definitions and Notation. E3={(xx, x2, x3) : xx, x2, x3 are real numbers}. E2={(xx, x2, x3) e A3 : x3=0}. Suppose A and A are point sets. Then : A + B denotes the union (sum) of A and A; Bd (A) denotes the set of boundary points of A ; Int (A) denotes the set of interior points of A; A denotes the closure of A. Suppose J is a simple closed curve in A2. Then In (J) denotes the bounded component (domain) of A2—J. Let A and A be metric spaces. Let fix and fi2 be continuous functions from A into A. Then: ||/i-/2| =sup {dist (fix(x),fi2(x)) : xeA}; fix= id means that fx(x) =x for all x e A ;fx : A -»- A means that/, is a continuous function from A onto A. A complementary domain of a planar set M is a component of A2 —M. A l-complex is a finite collection of arcs no two of which intersect in an interior point of either. A continuum is a closed, connected set. A continuous curve is a compact, locally connected, metric continuum. A nondegenerate cyclic element of a continuous curve M is a nondegenerate connected subset of M which contains no separating points and is maximal with respect to the property of being a connected subset without separating points. The fence over a planar set M is the set {(xx, x2, x3) e E3 : (xx, x2, 0) e A/}. An isotopy of a space M onto itself is a continuous function 77 from M x I onto M (where 7 is the interval [0, 1]) such that for each t0 e I, the function Hto, defined License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1969] HOMEOMORPHIC CONTINUOUS CURVES IN 2-SPACE 271 by 77(o(x)= 77(x, /0), is a homeomorphism of M onto M. A homeomorphism / from M onto M is realizable by an isotopy if there is an isotopy H of M onto M such that 77(x, 0)=x and H(x, l)=/(x) for all xeM. A finite or infinite sequence is a null sequence if the diameters of its elements converge to zero. If F is a bounded subset of E2 and C is a circle which encloses both D and its boundary, then the outer boundary of D is the boundary of the set of all points x such that x can be joined to some point of C by an arc which contains no point of D or of its boundary. By a theorem of R. L. Moore [8, Theorem 4, p. 259] if D is a bounded complementary domain of a continuous curve, then the outer bound- ary of F is a simple closed curve. A pinched annulus is any set homeomorphic to the set F defined as follows : Let Dx and D2 be planar disks such that Z>2<=Dx and Bd (7)2) n Bd (Dx) is a single point. Then F=C1 (Dx-D2). A separating point of a set M is a point/» e M such that M-{p} is not connected. An arc F is a spanning arc of a disk D if (a) B is contained in D, (b) the endpoints of B are contained in Bd (D), and (c) except for its endpoints, F misses Bd (D). An arc F is a spanning arc of an annulus A if (a) B is contained in A, (b) B inter- sects both boundary components of A, and (c) except for its endpoints, F misses Bd (A). §111 1. Purpose. In this section we shall prove the extension theorem for the case in which S contains no separating points. We shall make use of the following : Theorem 1 (R. L. Moore [9, p. 212]). Suppose X is a nondegenerate continuous curve in E2. Then the boundary of every domain of E2— X is a simple closed curve if and only if X contains no separating points. 2. Subdividing continuous curves which contain no separating points. Let S denote a nondegenerate planar continuous curve without separating points. Let 7 denote the boundary of the unbounded complementary domain of S. By Theorem 1, 7 is a simple closed curve. In proving the extension theorem it will be necessary to break up the interior (bounded complementary domain) of 7 into small pieces so that the boundaries of these pieces lie in S. This is the purpose of Theorem 2 and Theorem 5. Theorem 2. Let In (7) denote the bounded domain of E2—J. Let D be a bounded complementary domain of S. Then Cl(In(7)) = D+Rx + R2+---, where Rx, R2,... is a (finite or infinite) null sequence of disks, with disjoint interiors, such that, for each i, D n F( = 0 and Bd (Rt) is a subset of S. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 272 W. K. MASON [August Proof. By Theorem 1, the boundary of D is a simple closed curve. We shall consider three cases depending upon how Bd (D) intersects J. Case I. Bd (D) n J is more than one point. Diagram 1 In this case the theorem is obvious (see Diagram 1). Note that, for each /, Bd(A()c/+Bd(A- Case 2. Bd (D) n J is one point. Let/? be the point Bd (D) nj. S contains no separating points so there is an arc in S—{p} from Bd (A to J.